Geometric-Variational Approach for Color Image Enhancement and Segmentation

\Approximation of functionals depending on jumps by elliptic functionals via ?-convergence", Comm. \Existence theorem for a minimum problem with free discontinuity set", Arch. \Representation of colored images by manifolds embedded in higher dimensional non-Euclidean space", IEEE ICIP'98, Chicago, 1998. 10 one dimensional curves as expected. The reason is our numerical approximation for E. We use an edge image with the same resolution as that of the original image , adding central diierence approximation yield the edge regions. One possible solution is to apply the reened numerical approximation to the edge map as in 15]. Finally, in Figure 4 we apply the segmentation function E in the embedding space, to a noisy image. The source of the noise comes from a digital camera compression distortions, followed by a scanned version of a printout picture. Fig.4. The original noisy image is on the left, followed by the edge indicator eld E, and the nal result. Bottom line shows a zoom in on the original noisy on the left and ltered image. 6 Summary and Conclusions We presented a generalization of the Mumford-Shah enhancement and segmen-tation method. The generalization is in two aspects: Multi-channel images, i.e. color images are analyzed, and the L 2 measure is replaced by the Polyakov action. The generalization is a natural application of the Beltrami framework that represent images as an embedding map of the image manifold in a spatial-feature space. 9 Fig.3. Upper row, left to right: The original image, followed by the nal edge indicator function E, and the nal image. At the bottom are zoom-in frames of a square section cropped from the initial and the nal images. 8 5 Experimental results Fig.2. Upper row, left to right: The original noisy image, followed by the nal edge indicator function E, and the nal image. At the bottom are zoom-in frames of a square section cropped from the initial and nal images. We tested both cases, where the segmentation function E is deened on the image manifold and then on the embedding space. The time derivatives are approximated by an explicit forward numerical approximation (Euler scheme). The spatial derivatives were taken rst by forward followed by backwards approximation , see 17]. This is a simple way to keep the numerical support tight and centralized. The examples demonstrate color image enhancement for both noisy and clean images. In all examples we set = 710 ?2 ; = …